Elementary mathematical and computational tools for electrical and computer engineers using MATLAB

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ELEMENTARY MATHEMATICAL and COMPUTATIONAL TOOLS for ELECTRICAL and COMPUTER ENGINEERS USING MATLAB ®

© 2001 by CRC Press LLC

ELEMENTARY MATHEMATICAL and COMPUTATIONAL TOOLS for ELECTRICAL and COMPUTER ENGINEERS USING MATLAB ®

Jamal T. Manassah City College of New York

CRC Press Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Manassah, Jamal T. Elementary mathematical and computational tools for electrical and computer engineers using MATLAB/Jamal T. Manassah. p. cm. Includes bibliographical references and index. ISBN 0-8493-1080-6 1. Electrical engineering—Mathematics. 2. Computer science—Mathematics. 3. MATLAB. I. Title. TK153 .M362 2001 510′.24′62—dc21 2001016138

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2001 by CRC Press LLC No claim to original U.S. Government works International Standard Book Number 0-8493-1080-6 Library of Congress Card Number 2001016138 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper

About the Author

Jamal T. Manassah, has been Professor of Electrical Engineering at the City College of New York since 1981. He received his B.Sc. degree in Physics from the American University of Beirut, and his M.A. and Ph.D. in Theoretical Physics from Columbia University. Dr. Manassah was a Member of the Institute for Advanced Study. His current research interests are in theoretical and computational quantum and nonlinear optics, and in photonics.

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Introduction This book is mostly based on a series of notes for a primer course in electrical and computer engineering that I taught at the City College of New York School of Engineering. Each week, the class met for an hour of lecture and a three-hour computer laboratory session where students were divided into small groups of 12 to 15 students each. The students met in an informal learning community setting, a computer laboratory, where each student had the exclusive use of a PC. The small size of the groups permitted a great deal of individualized instruction, which was a key ingredient to cater successfully to the needs of students with heterogeneous high school backgrounds. A student usually takes this course in the second semester of his or her freshman year. Typically, the student would have completed one semester of college calculus, and would be enrolled in the second course of the college calculus sequence and in the first course of the physics sequence for students in the physical sciences and engineering. My purpose in developing this book is to help bring the beginner engineering student’s analytical and computational skills to a level of competency that would permit him or her to participate, enjoy, and succeed in subsequent electrical and computer engineering courses. My experience indicates that the lack of mastery of fundamental quantitative tools is the main impediment to a student’s progress in engineering studies. The specific goals of this book are 1. To make you more comfortable applying the mathematics and physics that you learned in high school or in college courses, through interactive activities. 2. To introduce you, through examples, to many new practical tools of mathematics, including discrete variables material that are essential to your success in future electrical engineering courses. 3. To instruct you in the use of a powerful computer program, MATLAB®*, which was designed to be simultaneously userfriendly and powerful in tackling efficiently the most demanding problems of engineering and sciences. 4. To give you, through the applications and examples covered, glimpses of some of the fascinating problems that an electrical or

* MATLAB® is a registered trademark of the MathWorks, Inc., 3 Apple Hill Drive, Natick, MA, 01760-2098, USA. Tel: 508-647-7000, Fax: 508-647-7101, e-mail: [email protected], Web: www.mathworks.com.

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computer engineer solves in the course of completing many of his or her design projects. My experience indicates that you can achieve the above goals through the following work habits that I usually recommend to my own students: • Read carefully the material from this book that is assigned to you by your instructor for the upcoming week, and make sure to solve the suggested preparatory exercises in advance of the weekly lecture. • Attend the lecture and follow closely the material presented, in particular the solutions to the more difficult preparatory exercises and the demonstrations. • Following the lecture, make a list of questions on the preparatory material to which you still seek answers, and ask your instructor for help and clarification on these questions, preferably in the first 30 minutes of your computer lab session. • Complete the in-class exercises during the computer lab session. If you have not finished solving all in-class exercises, make sure you complete them on your own, when the lab is open, or at home if you own a computer, and certainly before the next class session, along with the problems designated in the book as homework problems and assigned to you by your instructor. In managing this course, I found it helpful for both students and instructors to require each student to solve all problems in a bound notebook. The advantage to the student is to have easy access to his or her previous work, personal notes, and reminders that he or she made as the course progressed. The advantage to the instructor is to enhance his or her ability to assess, more easily and readily, an individual student’s progress as the semester progresses. This book may be used for self-study by readers with perhaps a little more mathematical maturity acquired through a second semester of college calculus. The advanced reader of this book who is familiar with numerical methods will note that, in some instances, I did not follow the canonical order for the sequence of presentation of certain algorithms, thus sacrificing some optimality in the structure of some of the elementary programs included. This was necessitated by the goal I set for this book, which is to introduce both analytical and computational tools simultaneously. The sections of this book that are marked with asterisks include material that I assigned as projects to students with either strong theoretical interest or more mathematical maturity than a typical second semester freshman student. Although incorporated in the text, they can be skipped in a first reading. I hope that, by their inclusion, I will facilitate to the interested reader a smooth transition to some new mathematical concepts and computational tools that are of particular interest to electrical engineers. © 2001 by CRC Press LLC

This text greatly benefited from course material previously prepared by my colleagues in the departments of electrical engineering and computer science at City College of the City University of New York, in particular, P. Combettes, I. Gladkova, B. Gross, and F. Thau. They provided either the starting point for my subsequent efforts in this course, or the peer critique for the early versions of this manuscript. I owe them many thanks and, of course, do not hold them responsible for any of the remaining imperfections in the text. The preparation of this book also owes a lot to my students. Their questions and interest in the material contributed to many modifications in the order and in the presentation of the different chapters. Their desire for working out more applications led me to expand the scope of the examples and exercises included in the text. To all of them, I am grateful. I am also grateful to Erwin Cohen, who introduced me to the fine team at CRC Press, and to Jerry Papke whose stewardship of the project from start to end at CRC Press was most supportive and pleasant. The editorial and production teams at CRC in particular, Samar Haddad, the project editor, deserve credit for the quality of the final product rendering. Naomi Fernandes and her colleagues at The MathWorks Inc. kindly provided me with a copy of the new release of MATLAB for which I am grateful. I dedicate this book to Azza, Tala, and Nigh whose support and love always made difficult tasks a lot easier. Jamal T. Manassah New York, January 2001

© 2001 by CRC Press LLC

Contents

1.

Introduction to MATLAB® and Its Graphics Capabilities 1.1 Getting Started 1.2 Basic Algebraic Operations and Functions 1.3 Plotting Points 1.3.1 Axes Commands 1.3.2 Labeling a Graph 1.3.3 Plotting a Point in 3-D 1.4 M-files 1.5 MATLAB Simple Programming 1.5.1 Iterative Loops 1.5.2 If-Else-End Structures 1.6 Array Operations 1.7 Curve and Surface Plotting 1.7.1 x-y Parametric Plot 1.7.2 More Parametric Plots in 2-D 1.7.3 Plotting a 3-D Curve 1.7.4 Plotting a 3-D Surface 1.8 Polar Plots 1.9 Animation 1.10 Histograms 1.11 Printing and Saving Work in MATLAB 1.12 MATLAB Commands Review

2.

Difference Equations 2.1 Simple Linear Forms 2.2 Amortization 2.3 An Iterative Geometric Construct: The Koch Curve 2.4 Solution of Linear Constant Coefficients Difference Equations 2.4.1 Homogeneous Solution 2.4.2 Particular Solution 2.4.3 General Solution 2.5 Convolution-Summation of a First-Order System with Constant Coefficients 2.6 General First-Order Linear Difference Equations* 2.7 Nonlinear Difference Equations 2.7.1 Computing Irrational Numbers 2.7.2 The Logistic Equation

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2.8

2.9

Fractals and Computer Art 2.8.1 Mira’s Model 2.8.2 Hénon’s Model Generation of Special Functions from Their Recursion Relations*

3.

Elementary Functions and Some of Their Uses 3.1 Function Files 3.2 Examples with Affine Functions 3.3 Examples with Quadratic Functions 3.4 Examples with Polynomial Functions 3.5 Examples with Trigonometric Functions 3.6 Examples with the Logarithmic Function 3.6.1 Ideal Coaxial Capacitor 3.6.2 The Decibel Scale 3.6.3 Entropy 3.7 Examples with the Exponential Function 3.8 Examples with the Hyperbolic Functions and Their Inverses 3.8.1 Capacitance of Two Parallel Wires 3.9 Commonly Used Signal Processing Functions 3.10 Animation of a Moving Rectangular Pulse 3.11 MATLAB Commands Review

4.

Numerical Differentiation, Integration, and Solutions of Ordinary Differential Equations 4.1 Limits of Indeterminate Forms 4.2 Derivative of a Function 4.3 Infinite Sums 4.4 Numerical Integration 4.5 A Better Numerical Differentiator 4.6 A Better Numerical Integrator: Simpson’s Rule 4.7 Numerical Solutions of Ordinary Differential Equations 4.7.1 First-Order Iterator 4.7.2 Higher-Order Iterators: The Runge-Kutta Method* 4.7.3 MATLAB ODE Solvers 4.8 MATLAB Commands Review

5.

Root Solving and Optimization Methods 5.1 Finding the Real Roots of a Function 5.1.1 Graphical Method 5.1.2 Numerical Methods 5.1.3 MATLAB fsolve and fzero Built-in Functions 5.2 Roots of a Polynomial

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5.3

5.4

Optimization Methods 5.3.1 Graphical Method 5.3.2 Numerical Methods 5.3.3 MATLAB fmin and fmins Built-in Function MATLAB Commands Review

6.

Complex Numbers 6.1 Introduction 6.2 The Basics 6.2.1 Addition 6.2.2 Multiplication by a Real or Imaginary Number 6.2.3 Multiplication of Two Complex Numbers 6.3 Complex Conjugation and Division 6.3.1 Division 6.4 Polar Form of Complex Numbers 6.4.1 New Insights into Multiplication and Division of Complex Numbers 6.5 Analytical Solutions of Constant Coefficients ODE 6.5.1 Transient Solutions 6.5.2 Steady-State Solutions 6.5.3 Applications to Circuit Analysis 6.6 Phasors 6.6.1 Phasor of Two Added Signals 6.7 Interference and Diffraction of Electromagnetic Waves 6.7.1 The Electromagnetic Wave 6.7.2 Addition of Electromagnetic Waves 6.7.3 Generalization to N-waves 6.8 Solving ac Circuits with Phasors: The Impedance Method 6.8.1 RLC Circuit Phasor Analysis 6.8.2 The Infinite LC Ladder 6.9 Transfer Function for a Difference Equation with Constant Coefficients* 6.10 MATLAB Commands Review

7.

Vectors 7.1 Vectors in Two Dimensions (2-D) 7.1.1 Addition 7.1.2 Multiplication of a Vector by a Real Number 7.1.3 Cartesian Representation 7.1.4 MATLAB Representation of the Above Results 7.2 Dot (or Scalar) Product 7.2.1 MATLAB Representation of the Dot Product 7.3 Components, Direction Cosines, and Projections 7.3.1 Components

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7.4

7.5

7.6 7.7 7.8 7.9 8.

7.3.2 Direction Cosines 7.3.3 Projections The Dirac Notation and Some General Theorems* 7.4.1 Cauchy-Schwartz Inequality 7.4.2 Triangle Inequality Cross Product and Scalar Triple Product* 7.5.1 Cross Product 7.5.2 Geometric Interpretation of the Cross Product 7.5.3 Scalar Triple Product Vector Valued Functions Line Integral Infinite Dimensional Vector Spaces* MATLAB Commands Review

Matrices 8.1 Setting up Matrices 8.1.1 Creating Matrices in MATLAB 8.2 Adding Matrices 8.3 Multiplying a Matrix by a Scalar 8.4 Multiplying Matrices 8.5 Inverse of a Matrix 8.6 Solving a System of Linear Equations 8.7 Application of Matrix Methods 8.7.1 dc Circuit Analysis 8.7.2 dc Circuit Design 8.7.3 ac Circuit Analysis 8.7.4 Accuracy of a Truncated Taylor Series 8.7.5 Reconstructing a Function from Its Fourier Components 8.7.6 Interpolating the Coefficients of an (n – 1)-degree Polynomial from n Points 8.7.7 Least-Square Fit of Data 8.8 Eigenvalues and Eigenvectors* 8.8.1 Finding the Eigenvalues of a Matrix 8.8.2 Finding the Eigenvalues and Eigenvectors Using MATLAB 8.9 The Cayley-Hamilton and Other Analytical Techniques* 8.9.1 Cayley-Hamilton Theorem dX = AX dt dX 8.9.3 Solution of Equations of the Form = AX + B(t ) dt 8.9.4 Pauli Spinors Special Classes of Matrices* 8.10.1 Hermitian Matrices 8.9.2

8.10

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Solution of Equations of the Form

8.11

8.10.2 Unitary Matrices 8.10.3 Unimodular Matrices MATLAB Commands Review

9.

Transformations 9.1 Two-dimensional (2-D) Geometric Transformations 9.1.1 Polygonal Figures Construction 9.1.2 Inversion about the Origin and Reflection about the Coordinate Axes 9.1.3 Rotation around the Origin 9.1.4 Scaling 9.1.5 Translation 9.2 Homogeneous Coordinates 9.3 Manipulation of 2-D Images 9.3.1 Geometrical Manipulation of Images 9.3.2 Digital Image Processing 9.3.3 Encrypting an Image 9.4 Lorentz Transformation* 9.4.1 Space-Time Coordinates 9.4.2 Addition Theorem for Velocities 9.5 MATLAB Commands Review

10.

A Taste of Probability Theory* 10.1 Introduction 10.2 Basics 10.3 Addition Laws for Probabilities 10.4 Conditional Probability 10.4.1 Total Probability and Bayes Theorems 10.5 Repeated Trials 10.5.1 Generalization of Bernoulli Trials 10.6 The Poisson and the Normal Distributions 10.6.1 The Poisson Distribution 10.6.2 The Normal Distribution

Supplement: Review of Elementary Functions S.1 Affine Functions S.2 Quadratic Functions S.3 Polynomial Functions S.4 Trigonometric Functions S.5 Inverse Trigonometric Functions S.6 The Natural Logarithmic Function S.7 The Exponential Function S.8 The Hyperbolic Functions S.9 The Inverse Hyperbolic Functions Appendix: Some Useful Formulae © 2001 by CRC Press LLC

Addendum: MATLAB 6 Selected References

*The asterisk indicates more advanced material that may be skipped in a first reading.

© 2001 by CRC Press LLC

1 Introduction to MATLAB® and Its Graphics Capabilities

1.1

Getting Started

MATLAB can be thought of as a library of programs that will prove very useful in solving many electrical engineering computational problems. MATLAB is an ideal tool for numerically assisting you in obtaining answers, which is a major goal of engineering analysis and design. This program is very useful in circuit analysis, device design, signal processing, filter design, control system analysis, antenna design, microwave engineering, photonics engineering, computer engineering, and all other sub-fields of electrical engineering. It is also a powerful graphic and visualization tool. The first step in using MATLAB is to know how to call it. It is important to remember that although the front-end and the interfacing for machines with different operating systems are sometimes different, once you are inside MATLAB, all programs and routines are written in the same manner. Only those few commands that are for file management and for interfacing with external devices such as printers may be different for different operating systems. After entering MATLAB, you should see the prompt >>, which means the program interpreter is waiting for you to enter instructions. (Remember to press the Return key at the end of each line that you enter.) Now type clf. This command creates a graph window (if one does not already exist) or clears an existing graph window. Because it is impossible to explain the function of every MATLAB command within this text, how would you get information on a certain command syntax? The MATLAB program has extensive help documentation available with simple commands. For example, if you wanted help on a function called roots (we will use this function often), you would type help roots. Note that the help facility cross-references other functions that may have related uses. This requires that you know the function name. If you want an idea of the available help files in MATLAB, type help. This gives you a list of topics included in MATLAB. To get help on a particular topic such as the Optimization Toolbox, type help toolbox/optim. This gives you a list of 0-8493-????-?/00/$0.00+$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC

all relevant functions pertaining to that area. Now you may type help for any function listed. For example, try help fmin.

1.2

Basic Algebraic Operations and Functions

The MATLAB environment can be used, on the most elementary level, as a tool to perform simple algebraic manipulations and function evaluations.

Example 1.1 Exploring the calculator functions of MATLAB. The purpose of this example is to show how to manually enter data and how to use basic MATLAB algebraic operations. Note that the statements will be executed immediately after they are typed and entered (no equal sign is required). Type and enter the text that follows the >> prompt to find out the MATLAB responses to the following: 2+2 5^2 2*sin(pi/4) The last command gave the sine of π/4. Note that the argument of the function was enclosed in parentheses directly following the name of the function. Therefore, if you wanted to find sin3(π/4), the proper MATLAB syntax would be sin(pi/4)^3 To facilitate its widespread use, MATLAB has all the standard elementary mathematical functions as built-in functions. Type help elfun, which is indexed in the main help menu to get a listing of some of these functions. Remember that this is just a small sampling of the available functions. help elfun The response to the last command will give you a large list of these elementary functions, some of which may be new to you, but all of which will be used in your future engineering studies, and explored in later chapters of this book.

Example 1.2 Assigning and calling values of parameters. In addition to inputting data directly to the screen, you can assign a symbolic constant or constants to rep© 2001 by CRC Press LLC

resent data and perform manipulations on them. For example, enter and note the answer to each of the following: a=2 b=3 c=a+b d=a*b e=a/b f=a^3/b^2 g=a+3*b^2 Question: From the above, can you deduce the order in which MATLAB performs the basic operations?

In-Class Exercise Pb. 1.1 Using the above values of a and b, find the values of: a. h = sin(a) sin(b) b. i = a1/3 b3/7 c. j = sin–1(a/b) = arcsin(a/b)

1.3

Plotting Points

In this chapter section, you will learn how to use some simple MATLAB graphics commands to plot points. We use these graphics commands later in the text for plotting functions and for visualizing their properties. To view all the functions connected with 2-dimensional graphics, type: help plot All graphics functions connected with 3-dimensional graphics can be looked up by typing help plot3 A point P in the x-y plane is specified by two coordinates. The x-coordinate measures the horizontal distance of the point from the y-axis, while the y-coordinate measures the vertical distance above the x-axis. These coordi-

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nates are called Cartesian coordinates, and any point in the plane can be described in this manner. We write for the point, P(x, y). Other representations can also be used to locate a point with respect to a particular set of axes. For example, in the polar representation, the point is specified by an r-coordinate that measures the distance of the point from the origin, while the θ-coordinate measures the angle which the line passing through the origin and this point makes with the x-axis. The purpose of the following two examples is to learn how to represent points in a plane and to plot them using MATLAB.

Example 1.3 Plot the point P(3, 4). Solution: Enter the following: x1=3; y1=4; plot(x1,y1,'*') Note that the semicolon is used in the above commands to suppress the echoing of the values of the inputs. The '*' is used to mark the point that we are plotting. Other authorized symbols for point displays include 'o', '+', 'x', … the use of which is detailed in help plot.

Example 1.4 Plot the second point, R(2.5, 4) on the graph while keeping point P of the previous example on the graph. Solution: If we went ahead, defined the coordinates of R, and attempted to plot the point R through the following commands: x2=2.5; y2=4; plot(x2,y2,'o') we would find that the last plot command erases the previous plot output. Thus, what should we do if we want both points plotted on the same graph? The answer is to use the hold on command after the first plot. The following illustrates the steps that you should have taken instead of the above: hold on x2=2.5;

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y2=4; plot(x2,y2,'o') hold off The hold off turns off the hold on feature. NOTES

1. There is no limit to the number of plot commands you can type before the hold is turned off. 2. An alternative method for viewing multiple points on the same graph is available: we may instead, following the entering of the values of x1, y1, x2, y2, enter: plot(x1,y1,'*',x2,y2,'o') This has the advantage, in MATLAB, of assigning automatically a different color to each point.

1.3.1

Axes Commands

You may have noticed that MATLAB automatically adjusts the scale on a graph to accommodate the coordinates of the points being plotted. The axis scaling can be manually enforced by using the command axis([xmin xmax ymin ymax]). Make sure that the minimum axis value is less than the maximum axis value or an error will result. In addition to being able to adjust the scale of a graph, you can also change the aspect ratio of the graphics window. This is useful when you wish to see the correct x to y scaling. For example, without this command, a circle will look more like an ellipse.

Example 1.5 Plot the vertices of a square, keeping the geometric proportions unaltered. Solution: Enter the following: x1=-1;y1=-1;x2=1;y2=-1;x3=-1;y3=1;x4=1;y4=1; plot(x1,y1,'o',x2,y2,'o',x3,y3,'o',x4,y4,'o') axis([-2 2 -2 2]) axis square %square shape Note that prior to the axis square command, the square looked like a rectangle. If you want to go back to the default aspect ratio, type axis normal. The % symbol is used so that you can type comments in your program. Comments following the % symbol are ignored by the MATLAB interpreter.

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1.3.2

Labeling a Graph

To add labels to your graph, the functions xlabel, ylabel, and title can be used as follows: xlabel('x-axis') ylabel('y-axis') title('points in a plane') If you desire to add a caption anywhere in the graph, you can use the MATLAB command gtext('caption') and place it at the location of your choice, on the graph, by clicking the mouse when the crosshair is properly centered there.

1.3.3

Plotting a Point in 3-D

In addition to being able to plot points on a plane (2-D space), MATLAB is also able to plot points in a three-dimensional space (3-D space). For this, we utilize the plot3 function.

Example 1.6 Plot the point P(3, 4, 5). Solution: Enter the following commands: x1=3; y1=4; z1=5; plot3(x1,y1,z1,'*') You can also plot multiple points in a 3-D space in exactly the same way as you did on a plane. Axis adjustment can still be used, but the vector input into the axis command must now have six entries, as follows: axis([xmin xmax ymin ymax zmin zmax]) You can similarly label your 3-D figure using xlabel, ylabel, zlabel, and title.

1.4

M-files

In the last section, we found that to complete a figure with a caption, we had to enter several commands one by one in the command window. Typing

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errors will be time-consuming to fix because if you are working in the command window, you need to retype all or part of the program. Even if you do not make any mistakes (!), all of your work may be lost if you inadvertently quit MATLAB and have not taken the necessary steps to save the contents of the important program that you just finished developing. To preserve large sets of commands, you can store them in a special type of file called an M-file. MATLAB supports two types of M-files: script and function M-files. To hold a large collection of commands, we use a script M-file. The function M-file is discussed in Chapter 3. To make a script M-file, you need to open a file using the built-in MATLAB editor. For both Macs and PCs, first select New from the file menu. Then select the M-file entry from the pull-down menu. After typing the M-file contents, you need to save the file: For Macs and PCs, select the save as command from the file window. A field will pop up in which you can type in the name you have chosen for this file (make sure that you do not name a file by a mathematical abbreviation, the name of a mathematical function, or a number). Also make sure that the file name has a .m extension added at the end of its name. For Macs, save the file in a user’s designated volume. For PCs, save the file in the default (bin) subdirectory. To run your script M-file, just type the filename (omitting the .m extension at its end) at the MATLAB prompt.

Example 1.7 For practice, go to your file edit window to create the following file that you name myfile.m. clear, clf x1=1;y1=.5;x2=2;y2=1.5;x3=3;y3=2; plot(x1,y1,'o',x2,y2,'+',x3,y3,'*') axis([0 4 0 4]) xlabel('xaxis') ylabel('yaxis') title('3points in a plane') After creating and saving myfile.m, go to the MATLAB command window and enter myfile. MATLAB will execute the instructions in the order of the statements stored in your myfile.m file.

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1.5

MATLAB Simple Programming

1.5.1

Iterative Loops

The power of computers lies in their ability to perform a large number of repetitive calculations. To do this without entering the value of a parameter or variable each time that these are changed, all computer languages have control structures that allow commands to be performed and controlled by counter variables, and MATLAB is no different. For example, the MATLAB “for” loop allows a statement or a group of statements to be repeated.

Example 1.8 Generate the square of the first ten integers. Solution: Edit and execute the the following script M-file: for m=1:10 x(m)=m^2; end; In this case, the number of repetitions is controlled by the index variable m, which takes on the values m = 1 through m = 10 in intervals of 1. Therefore, ten assignments were made. What the above loop is doing is sequentially assigning the different values of m^2 (i.e., m2) in each element of the “x-array.” An array is just a data structure that can hold multiple entries. An array can be 1-D such as in a vector, or 2-D such as in a matrix. More will be said about vectors and matrices in subsequent chapters. At this time, think of the 1-D and 2-D arrays as pigeonholes with numbers or ordered pair of numbers respectively assigned to them. To find the value of a particular slot of the array, such as slot 3, enter: x(3) To read all the values stored in the array, type: x Question: What do you get if you enter m? 1.5.2

If-Else-End Structures

If a sequence of commands must be conditionally evaluated based on a relational test, the programming of this logical relationship is executed with some variation of an if-else-end structure. © 2001 by CRC Press LLC

A. The simplest form of this structure is: if expression commands evaluated if expression is True else commands evaluated if expression is False end NOTES

1. The commands between the if and else statements are evaluated if all elements in the expression are true. 2. The conditional expression uses the Boolean logical symbols & (and), | (or), and ~ (not) to connect different propositions.

Example 1.9 Find for integer 0 < a ≤ 10, the values of C, defined as follows:  C= 3  2

ab for a > 5 ab for a ≤ 5

and b = 15. Solution: Edit and execute the following script M-file: for a=1:10 b=15; if a>5 C(a)=a*b; else C(a)=(a*b)*(3/2); end end Check that the values of C that you obtain by typing C are: 22.5 45 67.5 90 112.50 90 105 120 135 150 B. When there are three or more alternatives, the if-else-end structure takes the form: if expression 1 Commands 1 evaluated if expression 1 is True © 2001 by CRC Press LLC

elseif expression 2 Commands 2 evaluated if expression 2 is True elseif expression 3 Commands 3 evaluated if expression 3 is True … else Commands evaluated if no other expression is True end In this form, only the commands associated with the first True expression encountered are evaluated; ensuing relational expressions are not tested. 1.5.2.1 Alternative Syntax to the if Statement As an alternative to the if syntax, we can use, in certain instances, Boolean expressions to specify an expression in different domains. For example, (x>=l) has the value 1 if x is larger than or equal to 1 and zero otherwise; and (x=x1)

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Conditional evaluation. Gives the length of an array. Loads data or variable values from previous sessions into current MATLAB session. Generates an array with a specified number of points between two values. Makes a 2-D array of coordinate squares suitable for plotting surface meshes. Plots a mesh surface of a surface stored in a matrix. The same as mesh, but also plots in the same figure the contour plot. Finds the smallest element of an array. Finds the largest element of an array. Finds the mean of the elements of an array. Creates the matrix that contains the frames of an animation. Plays the movie described by a matrix M. Orients the current graph to your needs. Plots points or pairs of arrays on a 2-D graph. Plots points or array triples on a 3-D graph. Plots a polar plot on a polar grid. Polar to Cartesian conversion. Prints a figure to the default printer. Leave MATLAB program. Generates an array with elements randomly chosen from the uniform distribution over the interval [0, 1]. Generates an array with elements randomly chosen from the normal distribution function with zero mean and standard deviation 1. Partitions the graphics window into sub-windows. Saves MATLAB variables. Finds the standard deviation of the elements of an array. Plots the data sequence as stems from the x-axis terminated with circles for the data value. Views 3-D graphics from different perspectives. Lists all variables in the workspace. Labels the appropriate axes with text and title. Boolean function that is equal to 1 when the condition inside the parenthesis is satisfied, and zero otherwise.

2 Difference Equations This chapter introduces difference equations and examines some simple but important cases of their applications. We develop simple algorithms for their numerical solutions and apply these techniques to the solution of some problems of interest to the engineering professional. In particular, it illustrates each type of difference equation that is of widespread interest.

2.1

Simple Linear Forms

The following components are needed to define and solve a difference equation: 1. An ordered array defining an index for the sequence of elements 2. An equation connecting the value of an element having a certain index with the values of some of the elements having lower indices (the order of the equation being defined by the number of lower indices terms appearing in the difference equation) 3. A sufficient number of the values of the elements at the lowest indices to act as seeds in the recursive generation of the higher indexed elements. For example, the Fibonacci numbers are defined as follows: 1. The ordered array is the set of positive integers 2. The defining difference equation is of second order and is given by: F(k + 2) = F(k + 1) + F(k)

(2.1)

3. The initial conditions are F(1) = F(2) = 1 (note that the required number of initial conditions should be the same as the order of the equation). 0-8493-????-?/00/$0.00+$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC

From the above, it is then straightforward to compute the first few Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Example 2.1 Write a program for finding the first 20 Fibonacci numbers. Solution: The following program fulfills this task: N=18; F(1)=1; F(2)=1; for k=1:N F(k+2)=F(k)+F(k+1); end F It should be noted that the value of the different elements of the sequence depends on the values of the initial conditions, as illustrated in Pb. 2.1, which follows.

In-Class Exercises Pb. 2.1 Find the first 20 elements of the sequence that obeys the same recursion relation as that of the Fibonacci numbers, but with the following initial conditions: F(1) = 0.5

and F(2) = 1

Pb. 2.2 Find the first 20 elements of the sequence generated by the following difference equation: F(k + 3) = F(k) + F(k + 1) + F(k + 2) with the following boundary conditions: F(1) = 1, F(2) = 2,

and F(3) = 3

Why do we need to specify three initial conditions?

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2.2

Amortization

In this application of difference equations, we examine simple problems of finance that are of major importance to every engineer, on both the personal and professional levels. When the purchase of any capital equipment or real estate is made on credit, the assumed debt is normally paid for by means of a process known as amortization. Under this plan, a debt is repaid in a sequence of periodic payments where a portion of each payment reduces the outstanding principal, while the remaining portion is for interest on the loan. Suppose that the original debt to be paid is C and that interest charges are compounded at the rate r per payment period. Let y(k) be the outstanding principal after the kth payment, and u(k) the amount of the kth payment. After the kth payment period, the outstanding debt increased by the interest due on the previous principal y(k – 1), and decreased by the amount of payment u(k), this relation can be written in the following difference equation form: y(k) = (1 + r) y(k –1) – u(k)

(2.2)

We can simplify the problem and assume here that the bank wants its money back in equal amounts over N periods (this can be in days, weeks, months, or years; note, however, that whatever unit is used here should be the same as used for the assignment of the value of the interest rate r). Therefore, let u(k) = p

for k = 1, 2, 3, …, N

(2.3)

Now, using Eq. (2.2), let us iterate the first few terms of the difference equation: y(1) = (1 + r)y(0) – p = (1 + r)C – p

(2.4)

Since C is the original capital borrowed; At k = 2, using Eq. (2.2) and Eq. (2.4), we obtain: y(2) = (1 + r)y(1) – p = (1 + r)2C – p(1 + r) – p

(2.5)

At k = 3, using Eq. (2.2), (2.4), and (2.5), we obtain: y(3) = (1 + r)y(2) – p = (1 + r)3C – p(1 + r)2 – p(1 + r) – p etc. …

(2.6)

and for an arbitrary k, we can write, by induction, the general expression:

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k −1

y( k ) = (1 + r ) k C − p

∑ (1 + r)

i

(2.7)

i =0

Using the expression for the sum of a geometric series, from the appendix, the expression for y(k) then reduces to:  (1 + r ) k − 1  y( k ) = (1 + r ) k C − p   r  

(2.8)

At k = N, the debt is paid off and the bank is owed no further payment; therefore:  (1 + r ) N − 1  y( N ) = 0 = (1 + r ) N C − p   r  

(2.9)

From this equation, we can determine the amount of each of the (equal) payments: p=

r(1 + r ) N C (1 + r ) N − 1

(2.10)

Question: What percentage of the first payment is going into retiring the principal?

In-Class Exercises Pb. 2.3 Given the principal, the number of periods and the interest rate, use Eq. (2.10) to write a MATLAB program to find the amount of payment per period, assuming the payment per period is the same for all periods. Pb. 2.4 Use the same reasoning as for the amortization problem to write the difference equation for an individual’s savings plan. Let y(k) be the savings balance on the first day of the kth year and u(k) the amount of deposit made in the kth year. Write a MATLAB program to compute y(k) if the sequence u(k) and the interest rate r are given. Specialize to the case where you deposit an amount that increases by the rate of inflation i. Compute and plot the total value of the savings as a function of k if the deposit in the first year is $1000, the yearly interest rate is 6%, and the yearly rate of inflation is 3%. (Hint: For simplicity, assume that the deposits are made on December 31 of each year, and that the balance statement is issued on January 1 of each year.)

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FIGURE 2.1 The first few steps in the construction of the Koch curve.

2.3

An Iterative Geometric Construct: The Koch Curve

In your previous studies of 2-D geometry, you encountered classical geometric objects such as the circle, the triangle, the square, different polygons, etc. These shapes only approximate the shapes that you observe in nature (e.g., the shapes of clouds, mountain ranges, rivers, coastlines, etc.). In a successful effort to address the limitations of classical geometry, mathematicians have developed, over the last century and more intensely over the last three decades, a new geometry called fractal geometry. This geometry defines the geometrical object through an iterative transformation applied an infinite number of times on an initial simple geometrical object. We illustrate this new concept in geometry by considering the Koch curve (see Figure 2.1). The Koch curve has the following simple geometrical construction. Begin with a straight line of length L. This initial object is called the initiator. Now partition it into three equal parts. Then replace the middle line segment by an equilateral triangle (the segment you removed is its base). This completes the basic construction, which transformed the line segment into four non-colinear smaller parts. This constructional prescription is called the generator. We now repeat the transformation, taking each of the resulting line segments, partitioning them into three equal parts, removing the middle section, etc. © 2001 by CRC Press LLC

This process is repeated indefinitely. Figure 2.1 the first two steps of this construction. It is interesting to observe that the Koch curve is an example of a curve where there is no way to fit a tangent to any of its points. In a sense, it is an example of a curve that is made out of corners everywhere. The detailed study of these objects is covered in courses in fractal geometry, chaos, dynamic systems, etc. We limit ourselves here to the simple problems of determining the number of segments, the length of each segment, the length of the curve, and the area bounded by the curve and the horizontal axis, following the kth step: 1. After the first step, we are left with a curve made up of four line segments of equal length; after the second step, we have (4 × 4) segments; and the number of segments after k steps, is n(k) = 4k

(2.11)

2. If the initiator had length L, the length of the segment after the first step is L/3, L/(3)2, after the second step and after k steps: s(k) = L/(3)k

(2.12)

3. Combining the results of Eqs. (2.11) and (2.12), we deduce that the length of the curve after k steps:

P( k ) = L ×

 4  3

k

(2.13)

4. The number of vertices in this curve, denoted by u(k), is equal to the number of segments plus one: u(k) = 4k + 1

(2.14)

5. The area enclosed by the Koch curve and the horizontal line can be deduced from solving a difference equation: the area enclosed after the kth step is equal to the area enclosed in the (k – 1)th step plus the number of the added triangles multiplied by their individual area:

Number of new triangles =

 u( k ) − u( k − 1)    3

(2.15)

2k

Area of the new equilateral triangle =

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3 2 3  1 2 s (k ) = L (2.16) 4 4  3

from which the difference equation for the area can be deduced: 2  u( k ) − u( k − 1)  3 L A( k ) = A( k − 1) +   4 3 2 k 3 

3  2 = A( k − 1) + 24  3 

2 k −1

(2.17)

L2

The initial condition for this difference equation is: A(1) =

3 L2 4 9

(2.18)

Clearly, the solution of the above difference equation is the sum of a geometric series, and can therefore be written analytically. For k → ∞, this area has the limit: A( k → ∞) =

3 2 L 20

(2.19)

However, if you did not notice the relationship of the above difference equation with the sum of a geometric series, you can still solve this equation numerically, using the following routine and assuming L = 1: N=25; A=zeros(N,1); %preallocating size of array speeds % computation m=1:N; A(1)=(sqrt(3)/24)*(2/3); for k=2:N A(k)=A(k-1)+(sqrt(3)/24)*((2/3)^(2*k-1)); end stem(m,A,'*') The above plot shows the value of the area on the first 20 iterations of the function, and as can be verified, the numerical limit of this area has the same value as the analytical expression given in Eq. (2.19). Before leaving the Koch curve, we note that although the area of the curve goes to a finite limit as the index increases, the value of the length of the curve [Eq. (2.13)] continues to increase. This is a feature not encountered in the classical geometric objects with which you are most familiar.

© 2001 by CRC Press LLC

In-Class Exercise Pb. 2.5 Write a program to draw the Koch curve at the kth step. (Hint: Starting with the farthest left vertex and going clockwise, write a difference equation relating the coordinates of a vertex with those of the preceding vertex, the length of the segment, and the angle that the line connecting the two consecutive vertices makes with the x-axis.)

2.4

Solution of Linear Constant Coefficients Difference Equations

In Section 2.1, we explored the general numerical techniques for solving difference equations. In this section, we consider, some special techniques for obtaining the analytical solutions for the class of linear constant coefficients difference equations. The related physical problem is to determine, for a linear system, the output y(k), k > 0, given a specific input u(k) and a specific set of initial conditions. We discuss, at this stage, the so-called direct method. The general expression for this class of difference equation is given by: N

∑

M

a j y( k − j ) =

j =0

∑ b u(k − m) m

(2.20)

m= 0

The direct method assumes that the total solution of a linear difference equation is the sum of two parts — the homogeneous solution and the particular solution: y(k) = yhomog.(k) + ypartic.(k)

(2.21)

The homogeneous solution is independent of the input u(k), and the RHS of the difference equation is equated to zero; that is, N

∑ a y( k − j ) = 0 j

j =0

2.4.1 Homogeneous Solution Assume that the solution is of the form:

© 2001 by CRC Press LLC

(2.22)

yhomog.(k) = λk

(2.23)

Substituting in the homogeneous equation, we obtain the following algebraic equation: N

∑a λ

k− j

j

=0

(2.24)

j =0

or λk − N ( a0 λN + a1λN −1 + a2 λN −2 + … + aN −1λ + aN ) = 0

(2.25)

The polynomial in parentheses is called the characteristic polynomial of the system. The roots can be obtained analytically for all polynomials up to order 4; otherwise, they are obtained numerically. In MATLAB, they can be obtained graphically when they are all real, or through the roots command in the most general case. We introduce this command in Chapter 5. In all the following examples in this chapter, we restrict ourselves to cases for which the roots can be obtained analytically. If we assume that the roots are all distinct, the general solution to the homogeneous difference equation is: y homog. ( k ) = C1λk1 + C2 λk2 + … + CN λkN

(2.26)

where λ1, λ2, λ3, …, λN are the roots of the characteristic polynomial.

Example 2.2 Find the homogeneous solution of the difference equation y(k) – 3y(k – 1) – 4y(k – 2) = 0 Solution: The characteristic polynomial associated with this equation leads to the quadratic equation: λ2 – 3λ – 4 = 0 The roots of this equation are –1 and 4, respectively. Therefore, the solution of the homogeneous equation is: yhomog.(k) = C1(–1)k + C2(4)k The constants C1 and C2 are determined from the initial conditions y(1) and y(2). Substituting, we obtain:

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C1 = −

y( 2 ) y(1) + y(2) 4 y(1) + and C2 = 5 5 20

NOTE If the characteristic polynomial has roots of multiplicity m, then the portion of the homogeneous solution corresponding to that root can be written, instead of C1λk, as:

C1(1) λk + C1( 2 ) k λk + … + C1( m) k m−1λk

In-Class Exercises Pb. 2.6 Find the homogeneous solution of the following second-order difference equation: y(k) = 3y(k – 1) – 2y(k – 2) with the initial conditions: y(0) = 1 and y(1) = 2. Then check your results numerically. Pb. 2.7 Find the homogeneous solution of the following second-order difference equation: y(k) = [2 cos(θ)]y(k – 1) – y(k – 2) with the initial conditions: y(–2) = 0 and y(–1) = 1. Check your results numerically.

2.4.2 Particular Solution The particular solution depends on the form of the input signal. The following table summarizes the form of the particular solution of a linear equation for some simple input functions: Input Signal

Particular Solution

A (constant) AMk AkM {A cos(ω0k), A sin(ω0k)}

B (constant) BMk B0kM + B1kM–1 + … + BM B1 cos(ω0k) + B2 sin(ω0k)

For more complicated input signals, the z-transform technique provides the simplest solution method. This technique is discussed in great detail in courses on linear systems.

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In-Class Exercise Pb. 2.8 Find the particular solution of the following second-order difference equation: y(k) – 3y(k – 1) + 2y(k – 2) = (3)k

for k > 0

2.4.3 General Solution The general solution of a linear difference equation is the sum of its homogeneous solution and its particular solution, with the constants adjusted, so as to satisfy the initial conditions. We illustrate this general prescription with an example.

Example 2.3 Find the complete solution of the first-order difference equation: y(k + 1) + y(k) = k with the initial condition y(0) = 0. Solution: First, solve the homogeneous equation y(k + 1) + y(k) = 0. The characteristic polynomial is λ + 1 = 0; therefore, yhomog. = C(–1)k The particular solution can be obtained from the above table. Noting that the input signal has the functional form kM, with M = 1, then the particular solution is of the form: ypartic. = B0k + B1

(2.27)

Substituting back into the original equation, and grouping the different powers of k, we deduce that: B0 = 1/2

and B1 = –1/4

The complete solution of the difference equation is then: y( k ) = C( −1) k +

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2k − 1 4

The constant C is determined from the initial condition: y(0) = 0 = C( −1)0 +

( −1) 4

giving for the constant C the value 1/4.

In-Class Exercises Pb. 2.9 Use the following program to model Example 2.3: N=19; y(1)=0; for k=1:N y(k+1)=k-y(k); end y Verify the closed-form answer. Pb. 2.10 Find, for k ≥ 2, the general solution of the second-order difference equation: y(k) – 3y(k – 1) – 4y(k – 2) = 4k + 2 × 4k–1 with the initial conditions y(0) = 1 and y(1) = 9. (Hint: When the functional form of the homogeneous and particular solutions are the same, use the same functional form for the solutions as in the case of multiple roots for the characteristic polynomial.) 26 k  6 k   1 Answer: y( k ) = − ( −1) k + ( 4)  k 4  25  25  5

Homework Problems Pb. 2.11 Given the general geometric series y(k), where: y(k) = 1 + a + a2 + … + ak show that y(k) obeys the first-order equation:

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y(k) = y(k – 1) + ak Pb. 2.12 Show that the response of the system: y(k) = (1 – a)u(k) + a y(k – 1) to a step signal of amplitude c; that is, u(k) = c for all positive k, is given by: y(k) = c(1 – ak+1)

for k = 0, 1, 2, …

where the initial condition y(–1) = 0. Pb. 2.13 Given the first-order difference equation: y(k) = u(k) + y(k – 1)

for k = 0, 1, 2, …

with the input signal u(k) = k, and the initial condition y(–1) = 0. Verify that its solution also satisfies the second-order difference equation y(k) = 2y(k – 1) – y(k – 2) + 1 with the initial conditions y(0) = 0 and y(–1) = 0. Pb. 2.14 Verify that the response of the system governed by the first-order difference equation: y(k) = bu(k) + a y(k – 1) to the alternating input: u(k) = (–1)k for k = 0, 1, 2, 3, … is given by: y( k ) =

b [( −1) k + a k +1 ] for k = 0, 1, 2, 3, … 1+ a

if the initial condition is: y(–1) = 0. Pb. 2.15 The impulse response of a system is the output from this system when excited by an input signal δ(k) that is zero everywhere, except at k = 0, where it is equal to 1. Using this definition and the general form of the solution of a difference equation, write the output of a linear system described by: y(k) – 3y(k – 1) – 4y(k – 2) = δ(k) + 2δ(k – 1) The initial conditions are: y(–2) = y(–1) = 0. 6  1  Answer: y( k ) = − ( −1) k + ( 4) k  for k > 0 5  5 

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Pb. 2.16 The expression for the National Income is given by: y(k) = c(k) + i(k) + g(k) where c is consumer expenditure, i is the induced private investment, g is the government expenditure, and k is the accounting period, typically corresponding to a particular quarter. Samuelson theory, introduced to many engineers in Cadzow’s classic Discrete Time Systems (see reference list), assumes the following properties for the above three components of the National Income: 1. Consumer expenditure in any period k is proportional to the National Income at the previous period: c(k) = ay(k – 1) 2. Induced private investment in any period k is proportional to the increase in consumer expenditure from the preceding period: i(k) = b[c(k) – c(k – 1)] = ab[y(k – 1) – y(k – 2)] 3. Government expenditure is the same for all accounting periods: g(k) = g Combining the above equations, the National Income obeys the secondorder difference equation: y(k) = g + a(1 + b) y(k – 1) – aby(k – 2)

for k = 1, 2, 3, …

The initial conditions y(–1) and y(0) are to be specified. Plot the National Income for the first 40 quarters of a new national entity, assuming that: a = 1/6, b = 1, g = $10,000,000, y(–1) = $20,000,000, y(0) = $30,000,000. How would the National Income curve change if the marginal propensity to consume (i.e., the constant a) is decreased to 1/8?

2.5

Convolution-Summation of a First-Order System with Constant Coefficients

The amortization problem in Section 2.2 was solved by obtaining the present output, y(k), as a linear combination of the present and all past inputs, (u(k),

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u(k – 1), u(k – 2), …). This solution technique is referred to as the convolutionsummation representation: ∞

y( k ) =

∑ w(i) u(k − i)

(2.28)

i =0

where the w(i) is the weighting function (or weight). Usually, the infinite sum is reduced to a finite sum because the inputs with negative indexes are usually assumed to be zeros. On the other hand, in the difference equation formulation of this class of problems, the present output y(k) is expressed as a linear combination of the present and m most recent inputs and of the n most recent outputs, specifically: y(k) = b0u(k) + b1u(k – 1) + … + bmu(k – m) – a1y(k – 1) – a2y(k – 2) – … – any(k – n)

(2.29)

where, of course, n is the order of the difference equation. Elementary techniques for solving this class of equations were introduced in Section 2.4. However, the most powerful technique to directly solve the linear difference equation with constant coefficients is, as pointed out earlier, the z-transform technique. Each of the above formulations of the input-output problem has distinct advantages in different circumstances. The direct difference equation formulation is the most amenable to numerical computations because of lower computer memory requirements, while the convolution-summation technique has the advantage of being suitable for developing mathematical proofs and finding general features for the difference equation. Relating the parameters of the two formulations of this problem is usually cumbersome without the z-transform technique. However, for first-order difference equations, this task is rather simple.

Example 2.4 Relate, for a first-order difference equation with constant coefficients, the sets {an} and {bn} with {wn}. Solution: The first-order difference equation is given by: y(k) = b0u(k) + b1u(k – 1) – a1y(k – 1) where u(k) = 0 for all k negative. From the difference equation and the initial conditions, we can directly write: y(0) = b0u(0) © 2001 by CRC Press LLC

y(1) = b0 u(1) + b1u(0) − a1 y(0)  for k = 1,  = b0 u(1) + b1u(0) − a1b0 u(0)  = b0 u(1) + (b1 − a1b0 )u(0)  Similarly, y(2) = b0 u(2) + (b1 − a1b0 )u(1) − a1 (b1 − a1b0 )u(0) y(3) = b0 u(3) + (b1 − a1b0 )u(2) − a1 (b1 − a1b0 )u(1) + a12 (b1 − a1b0 )u(0) or, more generally, if: y(k) = w(0)u(k) + w(1)u(k – 1) + … + w(k)u(0) then, w(0) = b0 w(i) = ( − a1 )i −1 (b1 − a1b0 ) for i = 1, 2, 3, …

In-Class Exercises Pb. 2.17 Using the convolution-summation technique, find the closed form solution for: y( k ) = u( k ) −

1 1 u( k − 1) + y( k − 1) 3 2

u( k ) = 0 and the input function given by:  u( k ) = 1

for k negative otherwise

Compare your analytical answer with the numerical solution. Pb. 2.18 Show that the resultant weight functions for two systems are, respectively: w(k) = w1(k) + w2(k)

if connected in parallel

k

w( k ) =

∑ w (i ) w ( k − i ) 2

i =0

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1

if connected in cascade

2.6

General First-Order Linear Difference Equations*

Thus far, we have considered difference equations with constant coefficients. Now we consider first-order difference equations with arbitrary functions as coefficients: y(k + 1) + A(k)y(k) = B(k)

(2.30)

The homogeneous equation corresponding to this form satisfies the following equation: l(k + 1) + A(k)l(k) = 0

(2.31)

Its expression can be easily found: l( k + 1) = − A( k )l( k ) = A( k ) A( k − 1)l( k − 1) = … =  = ( −1) k +1 A( k ) A( k − 1)… A(0)l(0) =  

k



∏ [− A(i)]l(0)

(2.32)

i =0

Assuming that the general solution is of the form: y(k) = l(k)v(k)

(2.33)

let us find v(k). Substituting the above trial solution in the difference equation, we obtain: l(k + 1)v(k + 1) + A(k)l(k)v(k) = B(k)

(2.34)

Further, assuming that v(k + 1) = v(k) + ∆v(k)

(2.35)

substituting in the difference equation, and recalling that l(k) is the solution of the homogeneous equation, we obtain: ∆v( k ) =

B( k ) l( k + 1)

Summing this over the variable k from 0 to k, we deduce that: © 2001 by CRC Press LLC

(2.36)

k

v( k + 1) =

∑ l( j + 1) + C B( j )

(2.37)

j =0

where C is a constant.

Example 2.5 Find the general solution of the following first-order difference equation: y(k + 1) – k2 y(k) = 0 with y(1) = 1. Solution: y( k + 1) = k 2 y( k ) = k 2 ( k − 1) 2 y( k − 1) = k 2 ( k − 1)2 ( k − 2)2 y( k − 2) = k 2 ( k − 1)2 ( k − 2)2 ( k − 3)2 y( k − 3) = … = k 2 ( k − 1)2 ( k − 2)2 ( k − 3)2 …(2)2 (1)2 y(1) = ( k!)2

Example 2.6 Find the general solution of the following first-order difference equation: (k + 1)y(k + 1) – ky(k) = k2 with y(1) = 1. Solution: Reducing this equation to the standard form, we have: A( k ) = −

k k2 and B( k ) = k +1 k +1

The homogeneous solution is given by: l( k + 1) =

k! 1 = ( k + 1)! ( k + 1)

The particular solution is given by: k

v( k + 1) =

∑ j =1

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j2 ( j + 1) + C = ( j + 1)

k

∑j j =1

2

+C =

( k + 1)(2 k + 1)k +C 6

where we used the expression for the sum of the square of integers (see Appendix). The general solution is then: y( k + 1) =

(2 k + 1)k C + 6 ( k + 1)

From the initial condition y(1) = 1, we deduce that: C = 1.

In-Class Exercise Pb. 2.19 Find the general solutions for the following difference equations, assuming that y(1) = 1. a. y(k + 1) – 3ky(k) = 3k. b. y(k + 1) – ky(k) = k.

2.7

Nonlinear Difference Equations

In this and the following chapter section, we explore a number of nonlinear difference equations that exhibit some general features typical of certain classes of solutions and observe other instances with novel qualitative features. Our exploration is purely experimental, in the sense that we restrict our treatment to guided computer runs. The underlying theories of most of the models presented are the subject of more advanced courses; however, many educators, including this author, believe that there is virtue in exposing students qualitatively early on to these fascinating and generally new developments in mathematics. 2.7.1

Computing Irrational Numbers

In this model, we want to exhibit an example of a nonlinear difference equation whose solution is a sequence that approaches a specific limit, irrespective, within reasonable constraints, of the initial condition imposed on it. This type of difference equation has been used to compute a class of irrational numbers. For example, a well-defined approximation for computing A is the feedback process: y( k + 1) =

1 A  y( k ) +  2 y( k ) 

This equation’s main features are explored in the following exercise. © 2001 by CRC Press LLC

(2.38)

In-Class Exercise Pb. 2.20 Using the difference equation given by Eq. (2.38): a. Write down a routine to compute 2 . As an initial guess, take the initial value to be successively: 1, 1.5, 2; even consider 5, 10, and 20. What is the limit of each of the obtained sequences? b. How many iterations are required to obtain 2 accurate to four digits for each of the above initial conditions? c. Would any of the above properties be different for a different choice of A.

Now, having established that the above sequence goes to a limit, let us prove that this limit is indeed A . To prove the above assertion, let this limit be denoted by ylim; that is, for large k, both y(k) and y(k + 1) ⇒ ylim, and the above difference equation goes in the limit to: y lim =

1 A  y + 2  lim y lim 

(2.39)

Solving this equation, we obtain: y lim = A

(2.40)

It should be noted that the above derivation is meaningful only when a limit exists and is in the domain of definition of the sequence (in this case, the real numbers). In Section 2.7.2, we encounter a sequence where, for some values of the parameters, there is no limit.

2.7.2

The Logistic Equation

Section 2.7.1 illustrated the case in which the solution of a nonlinear difference equation converges to a single limit for large values of the iteration index. In this chapter subsection, we consider the case in which a succession of iterates (called orbits) bifurcate, yielding orbits of period length 2, 4, 8, 16, ad infinitum, ending in what is called a “chaotic” orbit of infinite period length. We illustrate the prototype for this class of difference equations by exploring the logistic difference equation. The logistic equation was introduced by Verhulst to model the growth of populations limited by finite resources (the name logistic was coined by the French army under Napoleon when this equation was used for the planning of “logement” of troops in camps). In more modern settings of ecology, the © 2001 by CRC Press LLC

above model is used to simulate a population growth model. Specifically, in an ecological or growth process, the normalized measure y(k + 1) of the next generation of a specie (the number of animals, for example) is a linear function of the present measure y(k); that is, y(k + 1) = ry(k)

(2.41)

where r is the growth parameter. If unchecked, the growth of the specie follows a geometric series, which for r > 1 grows to infinity. But growth is often limited by finite resources. In other words, the larger y(k), the smaller the growth factor. The simplest way to model this decline in the growth factor is to replace r by r(1 – y(k)), so that as y(k) approaches the theoretical limit (1 in this case), the effective growth factor goes to zero. The difference equation goes to: y(k + 1) = r(1 – y(k))y(k)

(2.42)

which is the standard form for the logistic equation. In the next series of exercises, we explore the solution of Eq. (2.42) as we vary the value of r. We find that qualitatively different classes of solutions may appear for different values of r. We start by writing the simple subroutine that models Eq. (2.42): N=127; r= ; y(1)= ; m=1:N+1; for k=1:N y(k+1)= r*(1-y(k))*y(k); end plot(m,y,'*') x The values of r and y(1) are to be keyed in for each of the specific cases under consideration.

In-Class Exercises In the following two problems, we take in the logistic equation r > 1 and y(1) < 1. Pb. 2.21 Consider the case that 1 < r < 3 and y(1) = 0.5. a. Show that by running the above program for different values of r and y(1) that the iteration of the logistic equation leads to the limit y( N >> 1) =

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 r − 1 .  r 

b. Does the value of this limit change if the value of y(1) is modified, while r is kept fixed? Pb. 2.22 Find the iterates of the logistic equation for the following values of r: 3.1, 3.236068, 3.3, 3.498561699, 3.566667, and 3.569946, assuming the following three initial conditions: y(1) = 0.2, y(1) = 0.5, y(1) = 0.7 In particular, specify for each case: a. The period of the orbit for large N, and the values of each of the iterates. b. Whether the orbit is super-stable (i.e., the periodicity is present for all values of N).

This section provided a quick glimpse of two types of nonlinear difference equations, one of which may not necessarily converge to one value. We discovered that a great number of classes of solutions may exist for different values of the equation’s parameters. In Section 2.8 we generalize to 2-D. Section 2.8 illustrates nonlinear difference equations in 2-D geometry. The study of these equations has led in the last few decades to various mathematical discoveries in the branches of mathematics called Symbolic Dynamical theory, Fractal Geometry, and Chaos theory, which have far-reaching implications in many fields of engineering. The interested student/reader is encouraged to consult the References section of this book for a deeper understanding of this subject.

2.8

Fractals and Computer Art

In Section 2.4, we introduced a fractal type having a priori well-defined and apparent spatial symmetries, namely, the Koch curve. In Section 2.7, we discovered that a certain type of 1-D nonlinear difference equation may lead, for a certain range of parameters, to a sequence that may have different orbits. Section 2.8.1 explores examples of 2-D fractals, generated by coupled difference equations, whose solution morphology can also be quite distinct due solely to a minor change in one of the parameters of the difference equations. Section 2.8.2 illustrates another possible feature observed in some types of fractals. We show how the 2-D orbit representing the solution of a particular nonlinear difference equation can also be substantially changed through a minor variation in the initial conditions of the equation.

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FIGURE 2.2 Plot of the Mira curve for a = 0.99. The starting point coordinates are (4, 0). Top panel: b = 1, bottom panel: b = 0.98.

2.8.1

Mira’s Model

The coordinates of the points on the Mira curve are generated iteratively through the following system of nonlinear difference equations: x( k + 1) = by( k ) + F( x( k )) y( k + 1) = − x( k ) + F(( x( k + 1)))

(2.43)

where

F( x) = ax +

2(1 − a)x 2 1 + x2

(2.44)

We illustrate the different morphologies of the solutions in two different cases, and leave other cases as exercises for your fun and exploration.

© 2001 by CRC Press LLC

Case 1 Here, a = –0.99, and we consider the cases b = 1 and b = 0.98. The starting point coordinates are (4, 0). See Figure 2.2. This case can be viewed by editing and executing the following script M-file: for n=1:12000 a=-0.99;b1=1;b2=0.98; x1(1)=4;y1(1)=0;x2(1)=4;y2(1)=0; x1(n+1)=b1*y1(n)+a*x1(n)+2*(1-a)*(x1(n))^2/(1+ (x1(n)^2)); y1(n+1)=-x1(n)+a*x1(n+1)+2*(1-a)*(x1(n+1)^2)/(1+ (x1(n+1)^2)); x2(n+1)=b2*y2(n)+a*x2(n)+2*(1-a)*(x2(n))^2/(1+ (x2(n)^2)); y2(n+1)=-x2(n)+a*x2(n+1)+2*(1-a)*(x2(n+1)^2)/(1+ (x2(n+1)^2)); end subplot(2,1,1); plot(x1,y1,'.') title('a=-0.99 b=1') subplot(2,1,2); plot(x2,y2,'.') title('a=-0.99 b=0.98') Case 2 Here, a = 0.7, and we consider the cases b = 1 and b = 0.9998. The starting point coordinates are (0, 12.1). See Figure 2.3.

In-Class Exercise Pb. 2.23 Manifest the computer artist inside yourself. Generate new geometrical morphologies, in Mira’s model, by new choices of the parameters (–1 < a < 1 and b ≈ 1) and of the starting point. You can start with: a −0.48 −0.25 0.1 0.5 0.99

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b1 1 1 1 1 1

b2 0.93 0.99 0.99 0.9998 0.9998

( x1 , y 1 ) ( 4 , 0) ( 3 , 0) ( 3 , 0) ( 3 , 0) (0, 12)

a=0.7 b=1 15 10 5 0 -5 -10 -15 -20

-15

-10

-5

0

5

10

15

5

10

15

a=0.7 b=0.9998 15 10 5 0 -5 -10 -15 -20

-15

-10

-5

0

20

FIGURE 2.3 Plot of the Mira curve for a = 0.7. The starting point coordinates are (0, 12.1). Top panel: b = 1, bottom panel: b = 0.9998.

2.8.2

Hénon’s Model

The coordinates of the Hénon’s orbits are generated iteratively through the following system of nonlinear difference equations: x( k + 1) = ax( k + 1) − b( y( k ) − ( x( k ))2 ) y( k + 1) = bx( k + 1) + a( y( k ) − ( x( k ))2 )

(2.45)

where a ≤ 1 and b = 1 − a 2 . Executing the following script M-file illustrates the possibility of generating two distinct orbits if the starting points of the iteration are slightly different (here, a = 0.24), and the starting points are slightly different from each other. The two cases initial point coordinates are given, respectively, by (0.5696, 0.1622) and (0.5650, 0.1650). See Figure 2.4. a=0.24; b=0.9708; © 2001 by CRC Press LLC

FIGURE 2.4 Plot of two Hénon orbits having the same a = 0.25 but different starting points. (o) corresponds to the orbit with starting point (0.5696, 0.1622), (x) corresponds to the orbit with starting point (0.5650, 0.1650).

x1(1)=0.5696;y1(1)=0.1622; x2(1)=0.5650;y2(1)=0.1650; for n=1:120 x1(n+1)=a*x1(n)-b*(y1(n)-(x1(n))^2); y1(n+1)=b*x1(n)+a*(y1(n)-(x1(n))^2); x2(n+1)=a*x2(n)-b*(y2(n)-(x2(n))^2); y2(n+1)=b*x2(n)+a*(y2(n)-(x2(n))^2); end plot(x1,y1,'ro',x2,y2,'bx') 2.8.2.1 Demonstration Different orbits for Hénon’s model can be plotted if different starting points are randomly chosen. Executing the following script M-file illustrates the a = 0.24 case, with random initial conditions. See Figure 2.5. a=0.24; b=sqrt(1-a^2); rx=rand(1,40); ry=rand(1,40); © 2001 by CRC Press LLC

FIGURE 2.5 Plot of multiple Hénon orbits having the same a = 0.25 but random starting points.

for n=1:1500 for m=1:40 x(1,m)=-0.99+2*rx(m); y(1,m)=-0.99+2*ry(m); x(n+1,m)=a*x(n,m)-b*(y(n,m)-(x(n,m))^2); y(n+1,m)=b*x(n,m)+a*(y(n,m)-(x(n,m))^2); end end plot(x,y,'r.') axis([-1 1 -1 1]) axis square

2.9

Generation of Special Functions from Their Recursion Relations*

In this section, we go back to more classical mathematics. We consider the case of the special functions of mathematical physics. In this case, we need to

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define the iterated quantities by two indices: the order of the function and the value of the argument of the function. In many electrical engineering problems, it is convenient to use a class of polynomials called the orthogonal polynomials. For example, in filter design, the set of Chebyshev polynomials are of particular interest. The Chebyshev polynomials can be defined through recursion relations, which are similar to difference equations and relate the value of a polynomial of a certain order at a particular point to the values of the polynomials of lower orders at the same point. These are defined through the following recursion relation: Tk(x) = 2xTk–1(x) – Tk–2(x)

(2.46)

Now, instead of giving two values for the initial conditions as we would have in difference equations, we need to give the explicit functions for two of the lower-order polynomials. For example, the first- and second-order Chebyshev polynomials are T1(x) = x

(2.47)

T2(x) = 2x2 – 1

(2.48)

Example 2.7 Plot over the interval 0 ≤ x ≤ 1, the fifth-order Chebyshev polynomial. Solution: The strategy to solve this problem is to build an array to represent the x-interval, and then use the difference equation routine to find the value of the Chebyshev polynomial at each value of the array, remembering that the indexing should always be a positive integer. The following program implements the above strategy: N=5; x1=1:101; x=(x1-1)/100; T(1,x1)=x; T(2,x1)=2*x.^2-1; for k=3:N T(k,x1)=2.*x.*T(k-1,x1)-T(k-2,x1); end y=T(N,x1); plot(x,y)

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In-Class Exercise Pb. 2.24 By comparing their plots, verify that the above definition for the Chebyshev polynomial gives the same graph as that obtained from the closed-form expression: TN(x) = cos(N cos–1(x))

for 0 ≤ x ≤ 1

In addition to the Chebyshev polynomials, you will encounter other orthogonal polynomials in your engineering studies. In particular, the solutions of a number of problems in electromagnetic theory and in quantum mechanics (QM) call on the Legendre, Hermite, Laguerre polynomials, etc. In the following exercises, we explore, in a preliminary manner, some of these polynomials. We also explore another important type of the special functions: the spherical Bessel function.

Homework Problems Pb. 2.25 Plot the function y defined, in each case: (m + 2)Pm+ 2 ( x) = (2m + 3)xPm+1 ( x) − (m + 1)Pm ( x) a. Legendre polynomials:

P1 ( x) = x and P2 ( x) =

1 (3x 2 − 1) 2

For 0 ≤ x ≤ 1, plot y = P5(x) These polynomials describe the electric field distribution from a nonspherical charge distribution.  H ( x) = 2 xH m+1 ( x) − 2(m + 1)H m ( x) b. Hermite polynomials:  m+ 2 2  H 1 ( x) = 2 x and H 2 ( x) = 4 x − 2 For 0 ≤ x ≤ 6, plot y = A5 H 5 ( x) exp( − x 2 / 2), where Am = (2 m m! π ) −1/2 The function y describes the QM wave-function of the harmonic oscillator. c. Laguerre polynomials: Lm+ 2 ( x) = [(3 + 2m + − x)Lm+1 ( x) − (m + 1)2 Lm ( x)]/ (m + 2)  2 L1 ( x) = 1 − x and L2 ( x) = (1 − 2 x + x / 2) For 0 ≤ x ≤ 6, plot y = exp(–x/2)L5(x) The Laguerre polynomials figure in the solutions of the QM problem of atoms and molecules.

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Pb. 2.26 The recursion relations can, in addition to defining orthogonal polynomials, also define some special functions of mathematical physics. For example, the spherical Bessel functions that play an important role in defining the modes of spherical cavities in electrodynamics and scattering amplitudes in both classical and quantum physics are defined through the following recursion relation: jm+ 2 ( x ) = With j1 ( x) =

 3 + 2m  j ( x) − jm ( x)  x  m +1

3 cos( x) sin( x) cos( x)  3 1 − and j2 ( x) =  3 −  sin( x) − 2 x x x x2 x

Plot j5(x) over the interval 0 < x < 15.

© 2001 by CRC Press LLC

3 Elementary Functions and Some of Their Uses The purpose of this chapter is to illustrate and build some practice in the use of elementary functions in selected basic electrical engineering problems. We also construct some simple signal functions that you will encounter in future engineering analysis and design problems. It is essential to review the Supplement at the end of this book in case you want to refresh your memory on the particular elementary functions covered in the different chapter sections. NOTE

3.1

Function Files

To analyze and graph functions using MATLAB, we have to be able to construct functions that can be called from within the MATLAB environment. In MATLAB, functions are made and stored in function M-files. We already used one kind of M-file (script file) to store various executable commands in a routine. Function M-files differ from script M-files in that they have designated input(s) and output(s). The following is an example of a function. Type and save the following function in a file named aline.m: function y=aline(x) % (x,y) is a point on a line that has slope 3 % and y-intercept -5 y=3*x-5; NOTES

1. The word function at the beginning of the file makes it a function rather than a script file. 2. The function name, aline, that appears in the first line of this file should match the name that we assign to this file name when saving it (i.e., aline.m). Having created a function M-file in your user volume, move to the command window to learn how to call this function. There are two basic ways to use a function file: 0-8493-????-?/00/$0.00+$.50 © 2000 by CRC Press LLC © 2001 by CRC Press LLC

1. To evaluate the function for a specified value x=x1, enter aline(x1) to get the function value at this point; that is, y1 = 3x1 – 5. 2. To plot y1 = 3x1 – 5 for a range of x values, say [–2, 7], enter: fplot('aline',[-2,7]) The above example illustrates a function with one input and one output. The construction of a function M-file of a function having n inputs and m outputs starts with: NOTE

function [y1,y2,...,ym]=funname(x1,x2,...,xn) Above, using a function M-file, we showed a method to plot the defined function aline on the interval (–2, 7) using the fplot command. An alternative method is, of course, to use arrays, in the manner specified in Chapter 1. Specifically, we could have plotted the 'aline' function in the following alternate method: x=-2:.01:7; y=3*x-5; plot(x,y) To compare the two methods, we note that: 1. plot requires a user-supplied x-array (abscissa points) and a constructed y-array (ordinate points), while fplot only requires the name of the function file, defined previously and stored in a function M-file and the endpoints of the interval. 2. The fplot automatically creates a sampled domain that is used to plot the function, taking into account the type of function being plotted and using enough points to make the display appear continuous. On the other hand, plot requires that you choose the array length yourself. Both methods, therefore, have their own advantages and it depends on the particular problem whether to use plot or fplot. We are now in position to explore the use of some of the most familiar functions.

3.2

Examples with Affine Functions

The equation of an affine function is given by: © 2001 by CRC Press LLC

y(x) = ax + b

(3.1)

In-Class Exercises Pb. 3.1 Generate four function M-files for the following four functions: y1 ( x ) = 3 x + 2 ; y 2 ( x ) = 3 x + 5; y 3 ( x ) = −

x x + 3; y 4 ( x ) = − + 4 3 3

Pb. 3.2 Sketch the functions of Pb. 3.1 on the interval –5 < x < 5. What can you say about the angle between each of the two lines’ pairs. (Did you remember to make your aspect ratio = 1?) Pb. 3.3 Read off the graphs the coordinates of the points of intersection of the lines in Pb. 3.1. (Become familiar with the use and syntax of the zoom and ginput commands for a more accurate reading of the coordinates of a point.) Pb. 3.4 Write a function M-file for the line passing through a given point and intersecting another given line at a given angle. tan( a) + tan(b)    Hint: tan( a + b) =   1 − tan( a) tan(b) 

Application to a Simple Circuit The purpose of this application is to show that: 1. The solution to a simple circuit problem can be viewed as the simultaneous solution of two affine equations, or, equivalently, as the intersection of two straight lines. 2. The variations in the circuit performance can be studied through a knowledge of the affine functions, relating the voltages and the current. Consider the simple circuit shown in Figure 3.1. In the terminology of the circuit engineer, the voltage source VS is called the input to the circuit, and the current I and the voltage V are called the circuit outputs. Thus, this is an example of a system with one input and two outputs. As you may have studied in high school physics courses, all of circuit analysis with resistors as elements can be accomplished using Kirchhoff’s current law, Kirchoff’s voltage law, and Ohm’s law. • Kirchoff’s voltage law: The sum of all voltage drops around a closed loop is balanced by the sum of all voltage sources around the same loop.

© 2001 by CRC Press LLC

+

R1

I Vs

R

V

_

FIGURE 3.1 A simple resistor circuit.

• Kirchoff’s current law: The algebraic sum of all currents entering (exiting) a circuit node must be zero. (Assign the + sign to those currents that are entering the node, and the – sign to those current exiting the node.) • Ohm’s law: The ratio of the voltage drop across a resistor to the current passing through the resistor is a constant, defined as the resistance of the element; that is, R =

V I

The quantities we are looking for include (1) the current I through the circuit, and (2) the voltage V across the load resistor R. Using Kirchoff’s voltage law and Ohm’s law for resistance R1, we obtain: Vs = V + V1 = V + IR1

(3.2)

while applying Ohm’s law for the load resistor gives: V = IR

(3.3)

These two equations can be rewritten in the form of affine functions of I as functions of V:

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L1 : I =

(Vs − V ) R1

(3.4)

L2 : I =

V R

(3.5)

If we know the value of Vs, R, and R1, then Eqs. (3.4) and (3.5) can be represented as lines drawn on a plane with ordinate I and abscissa V. Suppose we are interested in finding the value of the current I and the voltage V when R1 = 100Ω, R = 100Ω, and Vs = 5 V. To solve this problem graphically, we plot each of the L1 and L2 functions on the same graph and find their point of intersection. The functions L1 and L2 are programmed as follows: function I=L1(V) R1=100; R=100; Vs=5; I=(Vs-V)/R1; function I=L2(V) R1=100; R=100; Vs=5; I=V/R; Because the voltage V is smaller than the source potential, due to losses in the resistor, a suitable domain for V would be [0, 5]. We now plot the two lines on the same graph: fplot('L1',[0,5]) hold on fplot('L2',[0,5]) hold off

In-Class Exercise Pb. 3.5 Verify that the two lines L1 and L2 intersect at the point: (I = 0.025, V = 2.5).

In the above analysis, we had to declare the numerical values of the parameters R1 and R in the definition of each of the two functions. This can, at best, be tedious if you are dealing with more than two function M-files or two parameters; or worse, can lead to errors if you overlook changing the values of the parameters in any of the relevant function M-files when you decide to modify them. To avoid these types of problems, it is good practice to call all

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functions from a single script M-file and link the parameters’ values together so that you only need to edit the calling script M-file. To link the values of parameters to all functions in use, you can use the MATLAB global command. To see how this works, rewrite the above function M-files as follows: function I=L1(V) global R1 R Vs=5; I=(Vs-V)/R1; function I=L2(V) global R1 R Vs=5; I=V/R;

% global statement

% global statement

The calling script M-file now reads: global R1 R R1=100; R=100; V=0:.01:5; I1=L1(V); I2=L2(V); plot(V,I1,V,I2,'-')

%global statement %set global resistance values %set the voltage range %evaluate I1 %evaluate I2 %plot the two curves

In-Class Exercise Pb. 3.6 In the above script M-file, we used arrays and the plot command. Rewrite this script file such that you make use of the fplot command.

Further Consideration of Figure 3.1 Calculating the circuit values for fixed resistor values is important, but we can also ask about the behavior of the circuit as we vary the resistor values. Suppose we keep R1 = 100Ω and Vs = 5 V fixed, but vary the value that R can take. To this end, an analytic solution would be useful because it would give us the circuit responses for a range of values of the circuit parameters R1, R, Vs. However, a plot of the lines L1 and L2 for different values of R can also provide a great deal of qualitative information regarding how the simultaneous solution to L1 and L2 changes as the value of R changes.

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The following problem serves to give you a better qualitative idea as to how the circuit outputs vary as different values are chosen for the resistor R.

In-Class Exercise Pb. 3.7 This problem still refers to the circuit of Figure 3.1. a. Redraw the lines L1 and L2, using the previous values for the circuit parameters. b. Holding the graph for the case R = 100Ω, sketch L1 and L2 again for R = 50Ω and R = 500Ω. How do the values of the voltage and the current change as R increases; and decreases? c. Determine the largest values of the current and voltage that can exist in this circuit when R varies over non-negative values. d. The usual nomenclature for the circuit conditions is as follows: the circuit is called an open circuit when R = ∞, while it is called a short circuit when R = 0. What are the (V, I) solutions for these two cases? Can you generalize your statement?

Now, to validate the qualitative results obtained in Pb. 3.7, let us solve analytically the L1 and L2 system. Solving this system of two linear equations in two unknowns gives, for the current and the voltage, the following expressions:  R  V ( R) =   Vs  R + R1 

(3.6)

 1  I ( R) =   Vs  R + R1 

(3.7)

Note that the above analytic expressions for V and I are neither linear nor affine functions in the value of the resistance.

In-Class Exercise Pb. 3.8 This problem still refers to the circuit of Figure 3.1. a. Keeping the values of Vs and R1 fixed, sketch the functions V(R) and I(R) for this circuit, and verify that the solutions you found previously in Pbs. 3.7 and 3.8, for the various values of R, agree with those found here.

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b. Given that the power lost in a resistive element is the product of the voltage across the resistor multiplied by the current through the resistor, plot the power through the variable resistor as a function of R. c. Determine the value of R such that the power lost in this resistor is maximized. d. Find, in general, the relation between R and R1 that ensures that the power lost in the load resistance is maximized. (This general result is called Thevenin’s theorem.)

3.3

Examples with Quadratic Functions

A quadratic function is of the form: y(x) = ax2 + bx + c

(3.8)

Preparatory Exercises Pb. 3.9 Find the coordinates of the vertex of the parabola described by Eq. (3.8) as functions of the a, b, c parameters. Pb. 3.10 If a = 1, show that the quadratic Eq. (3.8) can be factored as: y(x) = (x – x+)(x – x–) where x± are the roots of the quadratic equation. Further, show that, for arbitrary a, the product of the roots is

c −b , and their sum is . a a

In-Class Exercises Pb. 3.11 Develop a function M-file that inputs the two real roots of a seconddegree equation and returns the value of this function for an arbitrary x. Is this function unique? Pb. 3.12 In your elementary mechanics course, you learned that the trajectory of a projectile in a gravitational field (oriented in the –y direction) with

© 2001 by CRC Press LLC

an initial velocity v0, x in the x-direction and v0, y in the y-direction satisfies the following parametric equations: x = v0 ,x t and y = −

1 2 gt + v0 , y t 2

where t is time and the origin of the axis was chosen to correspond to the position of the particle at t = 0 and g = 9.8 ms–2 a. By eliminating the time t, show that the projectile trajectory y(x) is a parabola. b. Noting that the components of the initial velocity can be written as function of the projectile initial speed and its angle of inclination: v0, y = v0 sin(φ)

and v0, x = v0 cos(φ)

show that, for a given initial speed, the maximum range for the projectile is achieved when the inclination angle of the initial velocity is 45°. c. Plot the range for a fixed inclination angle as a function of the initial speed.

3.4

Examples with Polynomial Functions

As pointed out in the Supplement, a polynomial function is an expression of the form: p( x) = an x n + an−1x n−1 + … + a1x + a0

(3.9)

where an ≠ 0 for an nth-degree polynomial. In MATLAB, we can represent the polynomial function as an array: p = [an an−1 … a0 ]

(3.10)

Example 3.1 You are given the array of coefficients of the polynomial. Write a function Mfile for this polynomial using array operations. Let p = [1 3 2 1 0 3]: Solution: function y=polfct(x) p=[1 3 2 1 0 3];

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L=length(p); v=x.^[(L-1):-1:0]; y=sum(p.*v);

In-Class Exercises Pb. 3.13 Show that, for the polynomial p defined by Eq. (3.9), the product of the roots is ( −1)n

a0 a , and the sum of the roots is − n−1 . an an

Pb. 3.14 Find graphically the real roots of the polynomial p = [1 1 0 3].

3.5

3

2

Examples with the Trigonometric Functions

A time-dependent cosine function of the form: x = a cos(ωt + φ)

(3.11)

appears often in many applications of electrical engineering: a is called the amplitude, ω the angular frequency, and φ the phase. Note that we do not have to have a separate discussion of the sine function because the sine function, as shown in the Supplement, differs from the cosine function by a constant phase. Therefore, by suitably changing only the value of the phase parameter, it is possible to transform the sine function into a cosine function. In the following example, we examine the period of the different powers of the cosine function; your preparatory task is to predict analytically the relationship between the periods of the two curves given in Example 3.2 and then verify your answer numerically.

Example 3.2 Plot simultaneously, x1(t) = cos3(t) and x2 = cos(t) on t ∈ [0, 6π]. Solution: To implement this task, edit and execute the following script M-file: t=0:.2:6*pi; a=1;w=1; x1=a*(cos(w*t))^3;

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% t-array % desired parameters % x1-array constructed

x2=a*cos(w*t); % x2-array constructed plot(t,x1,t,x2,'--')

In-Class Exercises Pb. 3.15 Determine the phase relation between the sine and cosine functions of the same argument. Pb. 3.16 The meaning of amplitude, angular frequency, and phase can be better understood using MATLAB to obtain graphs of the cosine function for a family of a values, ω values, and φ values. a. With ω = 1 and φ = π/3, plot the cosine curves corresponding to a = 1:0.1:2. b. With a = 1 and ω = 1, plot the cosine curves corresponding to φ = 0:π/10:π. c. With a = 1 and φ = π/4, plot the cosine curves corresponding to ω = 1:0.1:2.

Homework Problem Pb. 3.17 Find the period of the function obtained by summing the following three cosine functions: x1 = 3 cos(t / 3 + π / 3), x 2 = cos(t + π), x 3 =

1 3  cos (t + π) 2  3

Verify your result graphically.

3.6 3.6.1

Examples with the Logarithmic Function Ideal Coaxial Capacitor

An ideal capacitor can be loosely defined as two metallic plates separated by an insulator. If a potential is established between the plates, for example through the means of connecting the two plates to the different terminals of a battery, the plates will be charged by equal and opposite charges, with the battery serving as a pump to move the charges around. The capacitance of a © 2001 by CRC Press LLC

capacitor is defined as the ratio of the magnitude of the charge accumulated on either of the plates divided by the potential difference across the plates. Using the Gauss law of electrostatics, it can be shown that the capacitance per unit length of an infinitely long coaxial cable is: C 2πε = l ln(b / a)

(3.12)

where a and b are the radius of the internal and external conductors, respectively, and ε is the permittivity of the dielectric material sandwiched between the conductors. (The permittivity of vacuum is approximately ε0 = 8.85 × 10–12, while that of oil, polystyrene, glass, quartz, bakelite, and mica are, respectively, 2.1, 2.6, 4.5–10, 3.8–5, 5, and 5.4-6 larger.)

In-Class Exercise Pb. 3.18 Find the ratio of the capacitance of two coaxial cables with the same dielectric material for, respectively: b/a = 5 and 50.

3.6.2

The Decibel Scale

In the SI units used by electrical engineers, the unit of power is the Watt. However, in a number of applications, it is convenient to express the power as a ratio of its value to a reference value. Because the value of this ratio can vary over several orders of magnitude, it is often more convenient to represent this ratio on a logarithmic scale, called the decibel scale:  P  G[dB] = 10 log   Pref 

(3.13)

where the function log is the logarithm to base 10. The table below converts the power ratio to its value in decibels (dB):

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P/Pref (10n)

dB values (10 n)

4 2 1 0.5 0.25 0.1 10–3

6 3 0 –3 –6 –10 –30

In-Class Exercise Pb. 3.19 In a measurement of two power values, P1 and P2, it was determined that: G1 = 9 dB

and G2 = –11 dB

Using the above table, determine the value of the ratio P1/P2.

3.6.3

Entropy

Given a random variable X (such as the number of spots on the face of a thrown die) whose possible outcomes are x1, x2, x3, …, and such that the probability for each outcome is, respectively, p(x1), p(x2), p(x3), … then, the entropy for this system described by the outcome of one random variable is defined by: N

H (X ) = −

∑ p(x ) log (p(x )) i

2

i

(3.14)

i =1

where N is the number of possible outcomes, and the logarithm is to base 2. The entropy is a measure of the uncertainty in the value of the random variable. In Information Theory, it will be shown that the entropy, so defined, is the number of bits, on average, required to describe the random variable X.

In-Class Exercises Pb. 3.20 In each of the following cases, find the entropy: a. N = 32 and p( xi ) =

1 for all i 32

1 1 1 1 1 1 1 1  b. N = 8 and p =  , , , , , , ,   2 4 8 16 64 64 64 64   1 1 1 1 c. N = 4 and p =  , , ,  2 4 8 8 1 1 1  d. N = 4 and p =  , , , 0 2 4 4 

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Pb. 3.21 Assume that you have two dices (die), one red and the other blue. Tabulate all possible outcomes that you can obtain by throwing these die together. Now assume that all you care about is the sum of spots on the two die. Find the entropy of the outcome.

Homework Problem Pb. 3.22 A so-called A-law compander (compressor followed by an expander) uses a compressor that relates output to input voltages by: y=±

Ax 1 + log( A)

for x ≤ 1 / A

y=±

1 + log( A x ) 1 + log( A)

for

1 ≤ x ≤1 A

Here, the + sign applies when x is positive and the – sign when x is negative. x = vi/V and y = vo/V, where vi and vo are the input and output voltages. The range of allowable voltages is –V to V. The parameter A determines the degree of compression. For a value of A = 87.6, plot y vs. x in the interval [–1, 1].

3.7

Examples with the Exponential Function

Take a few minutes to review the section on the exponential function in the Supplement before proceeding further. (Recall that exp(1) = e.)

In-Class Exercises Pb. 3.23 Plot the function y(x) = (x13 + x9 + x5 + x2 + 1) exp(–4x) over the interval [0,10]. Pb. 3.24 Plot the function y(x) = cos(5x) exp(–x/2)) over the interval [0, 10]. Pb. 3.25 From the results of Pbs. 3.23 and 3.24, what can you deduce about the behavior of a function at infinity if one of its factors is an exponentially decreasing function of x, while the other factor is a polynomial or trigonomet-

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ric function of x? What modification to the curve is observed if the degree of the polynomial is increased?

Application to a Simple RC Circuit The solution giving the voltage across the capacitor in Figure 3.2 following the closing of the switch can be written in the following form:   t   t  Vc (t) = Vc (0) exp − + Vs 1 − exp −  RC   RC   

(3.15)

Vc(t) is called the time response of the RC circuit, or the circuit output resulting from the constant input Vs. The time constant RC of the circuit has the units of seconds and, as you will observe in the present analysis and other problems in subsequent chapters, its ratio to the characteristic time of a given input potential determines qualitatively the output of the system.

FIGURE 3.2 The circuit used in charging a capacitor.

In-Class Exercise Pb. 3.26 A circuit designer can produce outputs of various shapes by selecting specific values for the circuit time constant RC. In the following simulations, you can examine the influence of this time constant on the response of the circuit of Figure 3.2. Using Vc(0) = 3 volts, Vs = 10 volts (capacitor charging process), and RC = 1 s: a. Sketch a graph of Vc(t). What is the asymptotic value of the solution? How long does it take the capacitor voltage to reach the value of 9 volts? b. Produce an M-file that will plot several curves of Vc(t) corresponding to:

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(i) RC = 1 (ii) RC = 5 (iii) RC = 10 Which of these time constants results in the fastest approach of Vc(t) toward Vs? c. Repeat the above simulations for the case Vs = 0 (capacitor discharge)? d. What would you expect to occur if Vc(0) = Vs?

Homework Problem Pb. 3.27 The Fermi-Dirac distribution, which gives the average population of electrons in a state with energy ε, neglecting the electron spin for the moment, is given by: f(ε) =

1 exp[(ε − µ ) / Θ] + 1

where µ is the Fermi (or chemical) potential and Θ is proportional to the absolute (or Kelvin) temperature. a. Plot the function f(ε) as function of ε, for the following cases: (i) µ = 1 and Θ = 0.002 (ii) µ = 0.03 and Θ = 0.025 (iii) µ = 0.01 and Θ = 0.025 (iv) µ = 0.001 and Θ = 0.001 b. What is the value of f(ε) when ε = µ? c. Determine the condition under which we can approximate the Fermi-Dirac distribution function by: f(ε) ≈ exp[(µ – ε)/Θ]

3.8 3.8.1

Examples with the Hyperbolic Functions and Their Inverses Capacitance of Two Parallel Wires

The capacitance per unit length of two parallel wires, each of radius a and having their axis separated by distance D, is given by: © 2001 by CRC Press LLC

C = l

πε 0  D cosh −1  2a 

(3.16)

where ε0 is the permittivity of air (taken to be that of vacuum) = 8.854 × 10–12 Farad/m. Question: Write this expression in a different form using the logarithmic function.

In-Class Exercises Pb. 3.28 Find the capacitance per unit length of two wires of radii 1 cm separated by a distance of 1 m. Express your answer using the most appropriate of the following sub-units: mF = 10 −3 F (milli-Farad); nF = 10 −9 F (nano-Farad); fF = 10 −15 F (femto-Farad);

µF = 10 –6 F (micro-Farad); pF = 10 −12 F (pico-Farad); aF = 10 −18 F (atto-Farad);

Pb. 3.29 Assume that you have two capacitors, one consisting of a coaxial cable (radii a and b) and the other of two parallel wires, separated by the distance D. Further assume that the radius of the wires is equal to the radius of D b as a function of , a a if we desire the two geometrical configurations for the capacitor to end up the inner cylinder of the coaxial cable. Plot the ratio

ε   = 2.6. having the same value for the capacitance.  Take ε   0

3.9

Commonly Used Signal Processing Functions

In studying signals and systems, you will also encounter, inter alia, the following functions (or variation thereof), in addition to the functions discussed previously in this chapter: • Unit step function • Unit slope ramp function

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FIGURE 3.3 Various useful signal processing functions.

• • • •

Unit area rectangle pulse Unit slope right angle triangle function Equilateral triangle function Periodic traces

These functions are plotted in Figure 3.3, and the corresponding function Mfiles are (x is everywhere a scalar): A. Unit Step function function y=stepf(x) global astep if x